Problem 8:

A square plate  x  is at temperature .  At time  the temperature is increased to  along one of the four sides while being held at  along the other three sides, and heat then flows into the plate according to .  When does the temperature reach  at the center of the plate?

Solution:

This is a standard heat-equation problem with homogeneous boundary conditions.  It will be slightly more convenient to take the plate as [0,2] x [0,2], and the side with  will be taken as .  The solution is obtained as the sum of a steady-state solution v  with  so that , and a solution of the equation with homogeneous boundary conditions,  with  so that .

 > aj:= int(5/sinh(j*Pi)*sin(j*Pi*y/2),y = 0 .. 2) assuming j::posint;

 > bij:= -aj*int(sin(i*Pi*x/2)*sinh(j*Pi*x/2),x = 0 .. 2) assuming i::posint,j::posint;

 > bij:= simplify(evalc(convert(%,exp))) assuming i::posint,j::posint;

We want to evaluate this at x=1, y=1.  By symmetry it is obvious that v(1,1) = 5/4.

As for , the factor  for  (some preliminary investigation assured us that the answer we want should be somewhere between about 1/4 and 1) if .  So this approximation to  should be accurate to approximately 30 decimals when  :

 > U:= 5/4 + add(add(bij*sin(i*Pi*1/2)*sin(j*Pi*1/2)*exp(-Pi^2*(i^2+j^2)/4*t),j=1..floor(sqrt(116-i^2))),i=1..10);

Now to solve :

 > Digits:= 40: fsolve(U=1,t=1/4 .. 1);

This actually has 39 correct digits.